The Powerball Mega
Millions jackpot for Wednesday's drawing, after no one won on
Saturday, is up to $430 million as of 3:10 PM Wednesday.

That is a pretty huge chunk of money. However, as we saw with
last week's similarly large Mega Millions jackpot, which
someone did win on Saturday, taking a closer look at the
underlying math of the lottery shows that it's probably a bad
idea to buy a ticket.

Consider the expected value

When trying to evaluate the outcome of a risky, probabilistic
event like the lottery, one of the first things to look at
isexpected value.

The expected value of a randomly decided process is found by
taking all of the possible outcomes of the process, multiplying
each outcome by its probability, and adding all of these numbers
up. This gives us a long-run average value for our random
process.

Expected value is helpful for assessing gambling outcomes. If my
expected value for playing the game, based on the cost of playing
and the probabilities of winning different prizes, is positive,
then,in
the long run, the game will make me money. If expected value
is negative, then this game is a net loser for me.

Lotteries are a great example of this kind of probabilistic
process. In Powerball,
for each $2 ticket you buy, you choose five numbers between 1 and
69 (represented by white balls in the drawing) and one number
between 1 and 26 (the red "powerball" in the drawing). Prizes are
then given out based on how many of the player's numbers
match the numbers chosen in the drawing.

Match all five of the white balls between 1 and 69,
and the extra red powerball number between 1 and 26, and you win
the jackpot. After that, smaller prizes are given out for
matching some subset of those numbers.

The Powerballs website helpfully provides a list of the
odds and prizes for each of the possible outcomes. We
can use those probabilities and prize sizes to evaluate the
expected value of a $2 ticket. Take each prize, subtract the
price of our ticket, multiply the net return by the probability
of winning, and add all those values up to get our expected
value:

Already, we can see that a Powerball ticket isn't a great
investment: We end up with an expected value of -$0.21, making a
ticket a losing proposition.

Unfortunately, when we consider some other aspects of the
lottery, it gets far worse.

Annuity versus lump sum

Looking at just the headline prize is a vast oversimplification.

First, the headline $430 million grand prize is paid out as an
annuity, meaning that rather than getting the whole amount all at
once, you get the $430 million spread out in smaller — but still
multimillion-dollar — annual payments over 30 years. If you
choose instead to take the entire cash prize at one time, you get
much less money up front: The cash payout value at the time of
writing is $273.4 million.

The question of whether to take the annuity or the cash is
somewhat nuanced. ThePowerball websitesays
the annuity option's payments increase by 5% each year,
presumably keeping up with and somewhat exceeding inflation.

On the other hand, the state is investing the cash somewhat
conservatively, in a mix of various US government and agency
securities. It's quite possible, although risky, to get a larger
return on the cash sum if it's invested wisely.

Further, having more money today is frequently better than taking
in money over a long period of time, since a larger investment
today will accumulate compound interest more quickly than smaller
investments made over time. This is referred to as
thetime
value of money.

Taxes make things much worse

In addition to comparing the annuity with the lump sum, there's
also the big caveat of taxes. While state income taxes vary, it's
possible that combined state, federal, and, in some
jurisdictions, local taxes could take as much as half of the
money.

Factoring this in, if we're taking home only half of our
potential prizes, our expected-value calculations move deeper
into negative territory, making our Powerball investment an
increasingly bad idea. Here's what we get from taking the
annuity, after factoring in our estimated 50% in taxes. The new
expected value is now underwater, at -$0.94: